WEIGHTED HOLOMORPHIC BESOV SPACES AND THEIR BOUNDARY VALUES

We study weighted holomorphic Besov spaces and their boundary values. Under certain restrictions on the weighted function and parameters, we establish the equivalent norms for holomorphic functions in terms of their boundary functions. Some results about embedding and interpolation are also included.

BOUNDEDNESS OF CALDERN-ZYGMUND OPERATORS ON BESOV SPACES AND ITS APPLICATION

In this article, the author introduces a class of non-convolution Calder′on-Zygmund operators whose kernels are certain sums involving the products of Meyer wavelets and their convolutions. The boundedness on Besov spaces Bp^0 ,q（1 ≤p,q ≤∞） is also obtained. Moreover, as an application, the author gives a brief proof of the known result that Hrmander condition can ensure the boundedness of convolution-type Calder′on-Zygmund operators on Besov spaces B^p0 ,q（1 ≤p,q ≤∞）. However, the proof is quite different from the previous one.

T1 THEOREM FOR BESOV SPACES ON NONHOMOGENEOUS SPACES

Suppose μ is a Radon measure on R^d, which may be non doubling. The only condition assumed on μ is a growth condition, namely, there is a constant Co 〉 0 such that for all x ∈ supp（μ） and r 〉 0, μ（B（x, r）） ≤ Cor^n, where 0 〈 n ≤ d. We prove T1 theorem for non doubling measures with weak kernel conditions. Our approach yields new results for kernels satisfying weakened regularity conditions, while recovering previously known Tolsa＇s results. We also prove T1 theorem for Besov spaces on nonhomogeneous spaces with weak kernel conditions given in [7] .

GENERALIZED BESOV SPACES AND TRIEBEL-LIZORKIN SPACES

In this paper the classical Besov spaces B^sp.q and Triebel-Lizorkin spaces F^sp.q for s∈R are generalized in an isotropy way with the smoothness weights { ｜2j｜^α→ln }7=0. These generalized Besov spaces and Triebel-Lizorkin spaces, denoted by B^α→p.q and F^α→p.q for α^→ E Nk and k ∈N, respectively, keep many interesting properties, such as embedding theorems （with scales property for all smoothness weights）, lifting properties for all parameters 5, and duality for index 0 〈 p 〈∞ By constructing an example, it is shown that there are infinitely many generalized Besov spaces and generalized Triebel-Lizorkin spaces lying between B^sp.q and ∪t〉s B^tp.q, and between F^sp.q and ∪t〉s F^tp.q, respectively.

ON OSCILLATING KERNELS IN THE BESOV SPACE

In this paper we consider a convolution operator Tf=p.v.Ω*f with Ω(x)=K(x)×e^((r))λ >0.where K(x)is a weak Calderon-Zygmund kernel and h(x)is a real-valued differentiable function. We give a boundedness criterion for such an operator to map the Besov space B_1^(0.1)(R^n)into itself.

Gevrey Regularity and Time Decay of Fractional Porous Medium Equation in Critical Besov Spaces

In this paper, we show the existence and regularity of mild solutions depending on the small initial data in Besov spaces to the fractional porous medium equation. When 1 < α ≤ 2, we prove global well-posedness for initial data with 1 ≤ p < ∞, 1 ≤ q ≤ ∞, and analyticity of solutions with 1 < p < ∞, 1 ≤ q ≤ ∞. In particular, we also proved that when α = 1, both u and belong to . We solve this equation through the contraction mapping method based on Littlewood-Paley theory and Fourier multiplier. Furthermore, we can get time decay estimates of global solutions in Besov spaces, which is as t → ∞.

A DISCRETE CHARACTERIZATION OF BESOV SPACES

The characterization of isotropic Besov spaces for in terms of progressive differences of a function on dyadic points is obtained. Moreover, for withan analogous characterization of anisotropic Besov spaces is presented.

Convergence Rates of Density Estimation in Besov Spaces

The optimality of a density estimation on Besov spaces Bsr,q(R) for the Lp risk was established by Donoho, Johnstone, Kerkyacharian and Picard (“Density estimation by wavelet thresholding,” The Annals of Statistics, Vol. 24, No. 2, 1996, pp. 508-539.). To show the lower bound of optimal rates of convergence Rn(Bsr,q, p), they use Korostelev and Assouad lemmas. However, the conditions of those two lemmas are difficult to be verified. This paper aims to give another proof for that bound by using Fano’s Lemma, which looks a little simpler. In addition, our method can be used in many other statistical models for lower bounds of estimations.

A Characterization of Besov Spaces of Para-Accretive Type and Its Application

There are two folds in this article. One fold is to characterize the Besov spaces of para-accretive type , which reduces to the classical Besov spaces when the para-accretive function is constant, by using a discrete Calderón-type reproducing formula and Plancherel-P?lya-type inequality associated to a para-accretive function b in Rn. The other is to show that a generalized singular integral operator T with extends to be bounded from for and , where ε is the regularity exponent of the kernel of T.

Triebel-Lizorkin Spaces and Besov Spaces Associated with Different Homogeneities and Boundedness of Composition Operators

In this paper,the author introduces new Triebel-Lizorkin spaces and Besov spaces associated with different homogeneities and proves that the composition of two Calderón-Zygmund singular integral operators with different homogeneities is bounded on these new Triebel-Lizorkin spaces and Besov spaces.

Inhomogeneous Besov and Triebel-Lizorkin spaces associated with a para-accretive function and their applications

In this paper,using inhomogeneous Calderon’s reproducing formulas and the space of test functions associated with a para-accretive function,the inhomogeneous Besov and TriebelLizorkin spaces are established.As applications,pointwise multiplier theorems are also obtained.

齐型空间上加权Besov空间与Triebel-Lizorkin空间的Tb定理

【目的】齐型空间自然地包含了欧氏空间R^(n)、光滑紧Riemann流形及Lipschitz区域的边界等,拟在齐型空间上建立奇异积分算子在加权Besov空间与Triebel-Lizorkin空间上有界的Tb定理。【方法】通过离散Calderón再生公式和几乎正交估计建立加权Besov空间与加权Triebel-Lizorkin空间的Plancherel-P8lya特征刻画,以保证函数空间的范数独立于恒等逼近的选取。【结果】获得了齐型空间上Calderón-Zygmund奇异积分算子在加权Besov空间及Triebel-Lizorkin空间上有界的充分条件。【结论】将欧氏空间上的Calderón-Zygmund奇异积分理论延拓到更广的齐型空间上,为奇异积分算子在函数空间上有界提供了判定方法。

Besov型空间上的Carleson测度不等式

本文给出了Besov空间B_(p)和对角Besov空间B_(p.n)上具有导数形式的Carleson测度不等式.同时,我们也指出相关结果对p=∞的情形并不成立.

Singular Integrals and Weighted Triebel-Lizorkin and Besov Spaces of Arbitrary Number of Parameters

Though the theory of Triebel-Lizorkin and Besov spaces in one-parameter has been developed satisfactorily, not so much has been done for the multiparameter counterpart of such a theory. In this paper, we introduce the weighted Triebel-Lizorkin and Besov spaces with an arbitrary number of parameters and prove the boundedness of singular integral operators on these spaces using discrete Littlewood-Paley theory and Calderon＇s identity. This is inspired by the work of discrete Littlewood- Paley analysis with two parameters of implicit dilations associated with the flag singular integrals recently developed by Han and Lu [12]. Our approach of derivation of the boundedness of singular integrals on these spaces is substantially different from those used in the literature where atomic decomposition on the one-parameter Triebel-Lizorkin and Besov spaces played a crucial role. The discrete Littlewood-Paley analysis allows us to avoid using the atomic decomposition or deep Journe＇s covering lemma in multiparameter setting.

Function spaces of Besov-type and Triebel-Lizorkin-type——a survey

This article is devoted to presenting a recapitulative introduction for the theory of Besov-type and Triebel-Lizorkin-type spaces developed in recent years.

The Besov Space B_1^(0.1) on Domains

Let Ω be a bounded Lipschitz domain. Define B1,r0.1（Ω）={f∈L1（Ω）: there is an F∈ B10.1（Rn） such that F|Ω=f| and B1,z0.1（Ω）={f∈B10.1（Rn）: f=0 on Rn\}. In this paper, the authors establish the atomic decompositions of these spaces. As by-products, the authors obtained the regularity on these spaces of the solutions to the Dirichlet problem and the Neumann problem of the Laplace equation on R+n.

Some Applications of Besov Spaces on Fractals

Let F be a compact d-set in R^n with 0 〈 d ≤ n, which includes various kinds of fractals. The author establishes an embedding theorem for the Besov spaces Bpq^s（F） of Triebel and the Sobolev spaces W^1,P（F,d,μ） of Hajtasz when s 〉 1, 1 〈 p 〈∞ and 0 〈 q ≤ ∞. The author also gives some applications of the estimates of the entropy numbers in the estimates of the eigenvalues of some fractal pseudodifferential operators in the spaces Bpq^0（F） and Fpq^0（F）.

The Commutators of Fractional Integrals on Besov Spaces

In this paper we consider the commutators of fractional integrals

The Fejer-Riesz Inequality for the Besov Spaces

A Fejer Riesz inequality for the weighted Besov spaces Bp,q is given. Some characteriza- tions of functions in Bp.q in terms of their Taylor coefficients are obtained.

Regularity Condition of Solutions to the Quasi-geostrophic Equations in Besov Spaces with Negative Indices

With a Hǒlder type inequality in Besov spaces, we show that every strong solution θ（t, x） on （0, T ） of the dissipative quasi-geostrophic equations can be continued beyond T provided that ⊥θ（t, x） ∈L 2γ/γ-2δ （（0, T ）; B^δ-γ/2 ∞∞（R^2）） for 0 〈 δ 〈 γ/2 .